Properties

Label 726.4.a.f
Level $726$
Weight $4$
Character orbit 726.a
Self dual yes
Analytic conductor $42.835$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [726,4,Mod(1,726)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(726, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("726.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 726.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(42.8353866642\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{6} + 16 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{6} + 16 q^{7} + 8 q^{8} + 9 q^{9} + 12 q^{10} - 12 q^{12} - 38 q^{13} + 32 q^{14} - 18 q^{15} + 16 q^{16} + 126 q^{17} + 18 q^{18} - 20 q^{19} + 24 q^{20} - 48 q^{21} + 168 q^{23} - 24 q^{24} - 89 q^{25} - 76 q^{26} - 27 q^{27} + 64 q^{28} - 30 q^{29} - 36 q^{30} - 88 q^{31} + 32 q^{32} + 252 q^{34} + 96 q^{35} + 36 q^{36} + 254 q^{37} - 40 q^{38} + 114 q^{39} + 48 q^{40} - 42 q^{41} - 96 q^{42} + 52 q^{43} + 54 q^{45} + 336 q^{46} - 96 q^{47} - 48 q^{48} - 87 q^{49} - 178 q^{50} - 378 q^{51} - 152 q^{52} + 198 q^{53} - 54 q^{54} + 128 q^{56} + 60 q^{57} - 60 q^{58} - 660 q^{59} - 72 q^{60} + 538 q^{61} - 176 q^{62} + 144 q^{63} + 64 q^{64} - 228 q^{65} + 884 q^{67} + 504 q^{68} - 504 q^{69} + 192 q^{70} + 792 q^{71} + 72 q^{72} - 218 q^{73} + 508 q^{74} + 267 q^{75} - 80 q^{76} + 228 q^{78} + 520 q^{79} + 96 q^{80} + 81 q^{81} - 84 q^{82} + 492 q^{83} - 192 q^{84} + 756 q^{85} + 104 q^{86} + 90 q^{87} + 810 q^{89} + 108 q^{90} - 608 q^{91} + 672 q^{92} + 264 q^{93} - 192 q^{94} - 120 q^{95} - 96 q^{96} + 1154 q^{97} - 174 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 −3.00000 4.00000 6.00000 −6.00000 16.0000 8.00000 9.00000 12.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.4.a.f 1
3.b odd 2 1 2178.4.a.e 1
11.b odd 2 1 6.4.a.a 1
33.d even 2 1 18.4.a.a 1
44.c even 2 1 48.4.a.c 1
55.d odd 2 1 150.4.a.i 1
55.e even 4 2 150.4.c.d 2
77.b even 2 1 294.4.a.e 1
77.h odd 6 2 294.4.e.h 2
77.i even 6 2 294.4.e.g 2
88.b odd 2 1 192.4.a.i 1
88.g even 2 1 192.4.a.c 1
99.g even 6 2 162.4.c.c 2
99.h odd 6 2 162.4.c.f 2
132.d odd 2 1 144.4.a.c 1
143.d odd 2 1 1014.4.a.g 1
143.g even 4 2 1014.4.b.d 2
165.d even 2 1 450.4.a.h 1
165.l odd 4 2 450.4.c.e 2
176.i even 4 2 768.4.d.c 2
176.l odd 4 2 768.4.d.n 2
187.b odd 2 1 1734.4.a.d 1
209.d even 2 1 2166.4.a.i 1
220.g even 2 1 1200.4.a.b 1
220.i odd 4 2 1200.4.f.j 2
231.h odd 2 1 882.4.a.n 1
231.k odd 6 2 882.4.g.f 2
231.l even 6 2 882.4.g.i 2
264.m even 2 1 576.4.a.q 1
264.p odd 2 1 576.4.a.r 1
308.g odd 2 1 2352.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 11.b odd 2 1
18.4.a.a 1 33.d even 2 1
48.4.a.c 1 44.c even 2 1
144.4.a.c 1 132.d odd 2 1
150.4.a.i 1 55.d odd 2 1
150.4.c.d 2 55.e even 4 2
162.4.c.c 2 99.g even 6 2
162.4.c.f 2 99.h odd 6 2
192.4.a.c 1 88.g even 2 1
192.4.a.i 1 88.b odd 2 1
294.4.a.e 1 77.b even 2 1
294.4.e.g 2 77.i even 6 2
294.4.e.h 2 77.h odd 6 2
450.4.a.h 1 165.d even 2 1
450.4.c.e 2 165.l odd 4 2
576.4.a.q 1 264.m even 2 1
576.4.a.r 1 264.p odd 2 1
726.4.a.f 1 1.a even 1 1 trivial
768.4.d.c 2 176.i even 4 2
768.4.d.n 2 176.l odd 4 2
882.4.a.n 1 231.h odd 2 1
882.4.g.f 2 231.k odd 6 2
882.4.g.i 2 231.l even 6 2
1014.4.a.g 1 143.d odd 2 1
1014.4.b.d 2 143.g even 4 2
1200.4.a.b 1 220.g even 2 1
1200.4.f.j 2 220.i odd 4 2
1734.4.a.d 1 187.b odd 2 1
2166.4.a.i 1 209.d even 2 1
2178.4.a.e 1 3.b odd 2 1
2352.4.a.e 1 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(726))\):

\( T_{5} - 6 \) Copy content Toggle raw display
\( T_{7} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T - 16 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 38 \) Copy content Toggle raw display
$17$ \( T - 126 \) Copy content Toggle raw display
$19$ \( T + 20 \) Copy content Toggle raw display
$23$ \( T - 168 \) Copy content Toggle raw display
$29$ \( T + 30 \) Copy content Toggle raw display
$31$ \( T + 88 \) Copy content Toggle raw display
$37$ \( T - 254 \) Copy content Toggle raw display
$41$ \( T + 42 \) Copy content Toggle raw display
$43$ \( T - 52 \) Copy content Toggle raw display
$47$ \( T + 96 \) Copy content Toggle raw display
$53$ \( T - 198 \) Copy content Toggle raw display
$59$ \( T + 660 \) Copy content Toggle raw display
$61$ \( T - 538 \) Copy content Toggle raw display
$67$ \( T - 884 \) Copy content Toggle raw display
$71$ \( T - 792 \) Copy content Toggle raw display
$73$ \( T + 218 \) Copy content Toggle raw display
$79$ \( T - 520 \) Copy content Toggle raw display
$83$ \( T - 492 \) Copy content Toggle raw display
$89$ \( T - 810 \) Copy content Toggle raw display
$97$ \( T - 1154 \) Copy content Toggle raw display
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